"[…] mathematics is not just a symbolism, a set of conventions for the use of special, formal vocabularies, but is intimately connected with the structure of rational thought, with reasoning practices. [...] mathematics is not just a language, and of refusing the foundationalist move of trying to reduce mathematics to logic, instead seeing mathematics as providing rational frameworks for science, is to set science against a background of rational structures and rational methods which itself has a built-in dynamics. The rational framework of science is itself historically conditioned, for it changes with developments in mathematics." (Mary Tiles, "Bachelard: Science and Objectivity", 1984)
"Scientific laws give algorithms, or procedures, for determining how systems behave. The computer program is a medium in which the algorithms can be expressed and applied. Physical objects and mathematical structures can be represented as numbers and symbols in a computer, and a program can be written to manipulate them according to the algorithms. When the computer program is executed, it causes the numbers and symbols to be modified in the way specified by the scientific laws. It thereby allows the consequences of the laws to be deduced." (Stephen Wolfram, "Computer Software in Science and Mathematics", 1984)
"A
computer is an interpreted automatic formal system - that is to say, a
symbol-manipulating machine." (John Haugeland, "Artificial
intelligence: The very idea", 1985)
"We
who are heirs to three recent centuries of scientific development can hardly
imagine a state of mind in which many mathematical objects were regarded as
symbols of spiritual truths or episodes in sacred history. Yet, unless we make
this effort of imagination, a fraction of the history of mathematics is
incomprehensible.” (Philip J Davis & Rueben Hersh, “The Mathematical
Experience”, 1985)
"When a graph is constructed, quantitative and categorical information is encoded, chiefly through position, size, symbols, and color. When a person looks at a graph, the information is visually decoded by the person's visual system. A graphical method is successful only if the decoding process is effective. No matter how clever and how technologically impressive the encoding, it is a failure if the decoding process is a failure. Informed decisions about how to encode data can be achieved only through an understanding of the visual decoding process, which is called graphical perception." (William S Cleveland, "The Elements of Graphing Data", 1985)
"Artificial
intelligence is based on the assumption that the mind can be described as some
kind of formal system manipulating symbols that stand for things in the world.
Thus it doesn't matter what the brain is made of, or what it uses for tokens in
the great game of thinking. Using an equivalent set of tokens and rules, we can
do thinking with a digital computer, just as we can play chess using cups, salt
and pepper shakers, knives, forks, and spoons. Using the right software, one
system (the mind) can be mapped onto the other (the computer)." (George
Johnson, "Machinery of the Mind: Inside the New Science of Artificial Intelligence",
1986)
"Meaning does not reside in the mathematical symbols. It resides in the cloud of thought enveloping these symbols. It is conveyed in words; these assign meaning to the symbols." (Marvin Chester, "Primer of Quantum Mechanics", 1987)
"[…]
the chain of possible combinations of the encounter can be studied as such, as
an order which subsists in its rigor, independently of all subjectivity.
Through cybernetics, the symbol is embodied in the apparatus - with which it is
not to be confused, the apparatus being just its support. And it is embodied in
it in a literally trans-subjective way." (Jacques Lacan, 1988)
"Western
culture’s world-view appears to be dominated by material objects. […] One of
the ways mathematics has gained its power is through the activity of
objectivising the abstractions from reality. Through its symbols (letters,
numerals, figures) mathematics has taught people how to deal with abstract
entities, as if they were objects." (Alan J Bishop, "Mathematics
education in its cultural context", Educational Studies in Mathematics 19,
1988)
"People
who have a casual interest in mathematics may get the idea that a topologist is
a mathematical playboy who spends his time making Möbius bands and other
diverting topological models. If they were to open any recent textbook in
topology, they would be surprised. They would find page after page of symbols,
seldom relieved by a picture or diagram." (Martin Gardner,
"Hexaflexagons and Other Mathematical Diversions", 1988)
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